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Unformatted text preview: 10708 Graphical Models: Homework 2 Due October 11th, beginning of class September 27, 2006 Instructions : There are seven questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries regarding the question. The last problem involves coding, which should be done in MATLAB. Do not attach your code to the writeup. Instead, copy your implementation to /afs/andrew.cmu.edu/course/10/708/Submit/your_andrew_id/HW2 Refer to the web page for policies regarding collaboration, due dates, and extensions. 1 I-equivalence [10 pts] [Khalid] 1. Prove that two network structures G 1 and G 2 are I-equivalent if and only if the following two conditions hold: (a) The two graphs have the same set of trails, and (b) A trail is active in G 1 iff it is active in G 2 . 2. Let G 1 and G 2 be two graphs over X . Prove that if G 1 and G 2 have the same skeleton and the same set of v-structures then they are I-equivalent. (Hint: use the result from part 1) 2 Decomposable Scores [10 pts] [Ajit] Decomposable scoring functions are those where the score of a network given data D can be represented as the sum of scores of each node given its parents and the data: score( G : D ) = X i FamScore( X i | Pa G i : D ) 1 In greedy structure search we explore the space of structures by applying a local operator to an existing Bayes net. Examples of local operators include adding an edge, deleting an edge, and reversing an edge. In this question you will show that if the scoring function is decomposable, then computing the change in score caused by a local operator can be computed efficiently. 1. Prove proposition 15.4.5 (Koller &amp; Friedman p. 656) 2. Prove proposition 15.4.6 (Koller &amp; Friedman p. 656) 3 Learning Edge Directions [15 pts] [Ajit] In this question, we consider a simpler form of structure learning for BNs: Assume we have a skeleton and want to build a BN from it. For each edge, we want to either assign a direction to this edge or delete it from the graph. For this problem, you can assume you are using some decomposable score, FamScore( X i | Pa X i ). 1. Consider the skeleton X 1- - X 2- - X 3 , what are the possible BNs that we are con- sidering in this problem? What is the score of each of the graphs?...
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